Problem 873

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claims/: The 3 claim pages of Problem 873, one per claimant's result; the problem's standing derives from them.

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Statement. Let $A=\{a_1<a_2<\cdots\}\subseteq \mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that

$$[a_i,a_{i+1},\ldots,a_{i+k-1}] < X,$$

where the left-hand side is the least common multiple. Is it true that, for every $\epsilon >0$, there exists some $k$ such that

$$F(A,X,k)<X^\epsilon?$$

Status. Open. The site's label is OPEN. Three pending partial claims answer the question yes for a range of exponents: the bound $F(A,X,3)\ll X^{1/3}\log X$ that Erdős reports with Szemerédi in 1992 (claim page), Ritvik Nayak's note of April 2026 with exponents below $1/3$ for $k=6$ and $k=11$ (claim page), and two notes posted by old-bielefelder in April and August 2026, written by ChatGPT 5.5 Thinking and ChatGPT 5.6 Sol, covering every exponent above $1/4$ (claim page). No claim covers exponents at most $1/4$, and the standing in the frontmatter is derived from the claim pages.

Kenta Kitamura's Lean development (thread post of 11 September 2026; developed with ChatGPT and OpenAI Codex using GPT-6 (Astra), as the post discloses) proves that every increasing sequence has $\liminf_{X\to\infty}F(A,X,3)/X^{1/3}\le6$. This refutes Erdős's suggestion in [Er92c], p. 48, repeated in the site's commentary, that some sequence satisfies $F(A,X,3)\gg X^{1/3}\log X$ for every $X$. Formal-conjectures registers the development as the formal proof of its variant erdos_873.variants.supplement_all_scale. It settles no instance of the question asked, so it has no claim page; it was not built or audited here.

Source. erdosproblems.com/873, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #873, https://www.erdosproblems.com/873.

References.

• [Er92c] Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. 15 (1992), 34–50. Library home: erdos_1992_my_forgotten_problems_number_theory.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• letendre_2025_divisors_integer_short_interval
• letendre_2025_divisors_integer_short_interval / proposition_1